Yes, many serial killers desire fame.

Yes, many serial killers desire fame, either consciously or unconsciously. The killer often views himself as an "artist" of his macabre craft. While realizing the rational need to stay under the radar to continue practicing his "art," all artists want an audience to admire the results. Thus, leaving breadcrumb trail for the police to follow may be completely unconscious.

Yes, serial killers not only want to be caught, their need to be caught may not be so secret.

Since 'Silence of the Lambs,' serials killers have learned how much more powerful they can feel when they play for the limelight. They need their deeds to be public. As well, some need to play chicken with the police. Since many do kill because of compulsions they cannot control, they also need help to stop. Once law enforcement learn to read the behavioral codes of a particular serial killer, they know more what kind of need the killer hunts from and when he or she starts asking for help.

Most serial killers want to kill

We have no idea how many serial killers there may be in the world, if they are smart enough to elude the police. It's the ones that stay in one area, kill too often, or are flat-out crazy that end up caught. History shows that successful serial killers have existed: Jack the Ripper wasn't caught.

Some of them, maybe, but generally no

Some serial killers get off on playing games with the police and dropping clues, but these seem to be the minority, although they make for a really good story and therefore get a lot of press. Most serial killers just enjoy killing and want to be free to do it as long as they can.